Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 897–925 | Cite as

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

Article

Abstract

A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.

Keywords

Reaction-diffusion equation Logistic equation Delayed and instantaneous density dependence Stability Local Hopf bifurcation Global Hopf bifurcation 

Mathematics Subject Classification (2010)

35K57 35R10 35B32 35B10 92D25 92D40 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Department of MathematicsCollege of William and MaryWilliamsburgUSA

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